Theorem 3.5 ([1]}) 
For every \(1 \le i \le n\) , after receiving \(P_i\) , Algorithm \(\mathcal {A}(k,n,\delta )\)  defines a function \(f_i\)  such that with probability \(1-\delta \) , using the above definition of \(\pi _i\) , the bound \(\sum _{p \in P_i} D(p,\pi _i(p)) \le \alpha \text{\footnotesize \textsf {OPT}}_k(P_i)\)  holds. The algorithm deterministically requires the storage of \(O(k (\log n + \log (1/\delta )))\)  points.

